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In this paper, a fast and robust self-representation (FRSR) method is proposed to select a proper band subset from hyperspectral imagery (HSI). The FRSR assumes the separability structure of the HSI band set and transforms the problem of separable nonnegative matrix factorization into the robust self-representation (RSR) model. Then, the FRSR incorporates structured random projections into the RSR model to improve computational efficiency. The solution of FRSR is formulated into optimizing a convex problem and the augmented Lagrangian multipliers are adopted to estimate the proper factorization localizing matrix in the FRSR. The selected band subset is constituted with the bands corresponding to the r largest diagonal entries of the factorization localizing matrix. The experimental results show that FRSR outperforms state-of-the-art techniques in classification accuracy with lower computational cost.